Objectif Dual parametric and implicit representations are central in algorithms for low degree (typically 1 or 2) algebraic surfaces. Although available in theory, exact implicit representations of sculptured CAD-type surfaces are not useful in practice, as they are computationally too expensive. This is due to the high degrees and the exploding number of coefficients. Since the advent of approximate implicitisation, new methods are within reach. A previous FET assessment project verified that self-intersection algorithms work with approximate implicit of total degree as low as 4. Intersection algorithms are the most complex part of 3D CAD kernels, and the source of many problems in advanced CAD model exchange and use. Approximate implicitisation opens a bridge to algebraic geometry. By combining results from different branches of mathematics, we plan to investigate the use of approximate algebraic geometry in surface intersection algorithms.DESCRIPTION OF WORK1. Singularities: Classification, detection and localisation. The partners believe that Computer Aided Geometric Design will benefit from algebraic geometry. The project will address both real and classical algebraic geometry to achieve a better understanding of intersection problems and identify results that can improve intersection algorithms.2. Representation: Different methods for finding algebraic representations exist. We will examine in more detail the resultant types, further investigate approximate methods and address piecewise algebraic methods. Topics of special interest are approximation with parametrical implicit curves/ surfaces and classes of algebraic surfaces governed by a small numbers of parameters.3. Intersection: As the main objective is intersection algorithms, we will examine how the approximate algebraic approach complements state-of-the-art methods, and develop prototype integration. More accurate methods and compact representations for intersection tracks will also be in focus. The results will be demonstrated in an industrial prototype.4. Applications: The assessment project resulted in a number of open issues that will be addressed. Some applications have already been identified, such as detection of loops on curves and ray tracing for graphics; others are foreseen. When a complex geometric constellation arises, a local approximate algebraic geometry is a tool to gain more insight and sort out the solution. As the project will benefit from industrial feedback from a larger industrial audience, the GAIA users club will invite industrial companies interested in influencing new CAD-type technology. Champ scientifique natural sciencesmathematicspure mathematicsgeometrynatural sciencesmathematicspure mathematicsalgebraalgebraic geometry Programme(s) FP5-IST - Programme for research, technological development and demonstration on a "User-friendly information society, 1998-2002" Thème(s) 1.1.2.-6.1.1 - FET O: Open domain Appel à propositions Data not available Régime de financement CSC - Cost-sharing contracts Coordinateur SINTEF - STIFTELSEN FOR INDUSTRIELL OG TEKNISK FORSKNING VED NORGES TEKNISKE HOEGSKOLE Contribution de l’UE Aucune donnée Adresse STRINDVEIEN 4 7034 TRONDHEIM Norvège Voir sur la carte Coût total Aucune donnée Participants (6) Trier par ordre alphabétique Trier par contribution de l’UE Tout développer Tout réduire CENTRE NATIONAL DE LA RECHERCHE SCIENTIFIQUE France Contribution de l’UE Aucune donnée Adresse 3, RUE MICHEL-ANGE 75794 PARIS CEDEX 16 Voir sur la carte Coût total Aucune donnée JOHANNES KEPLER UNIVERSITAET LINZ - INSTITUT FUER ANALYSIS Autriche Contribution de l’UE Aucune donnée Adresse ALTENBERGERSTRASSE 69 4040 LINZ Voir sur la carte Coût total Aucune donnée THINK3 France Contribution de l’UE Aucune donnée Adresse 55 AVENUE GALLINE 69100 VILLEURBANNE Voir sur la carte Coût total Aucune donnée UNIVERSIDAD DE CANTABRIA Espagne Contribution de l’UE Aucune donnée Adresse AVENIDA DE LOS CASTROS S/N 39005 CANTABRIA Voir sur la carte Coût total Aucune donnée UNIVERSITE DE NICE - SOPHIA ANTIPOLIS France Contribution de l’UE Aucune donnée Adresse 28 AV. DE VALROSE - PARC VALROSE 06108 NICE - CEDEX 2 Voir sur la carte Coût total Aucune donnée UNIVERSITETET I OSLO Norvège Contribution de l’UE Aucune donnée Adresse PROBLEMVEIEN 5-7 0316 OSLO Voir sur la carte Coût total Aucune donnée